Notes: Surface Area of Right Prisms and Cylinders

Prisms and cylinders have 2 congruent parallel bases.A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles.

A. Definitions and Vocabulary

An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude. For a right prism, the altitude is the same as the length of a lateral side.

Surface area is the total area of all faces and curvedsurfaces of a three-dimensional figure. The lateralarea of a prism is the sum of the areas of the lateral faces.

The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism.

The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh.

II. Lateral and Surface Area of Right Prisms

Ex 1a. Find the lateral area and surface area of the right rectangular prism. Round to the nearest tenth, if necessary.

L = Ph

= 32(14) = 448 ft2

S = Ph + 2B

= 448 + 2(7)(9) = 574 ft2

P = 2(9) + 2(7) = 32 ft

Ex 1b. Find the lateral area and surface area of a right regular triangular prism with height 20 cm , base edges of

length 10 cm, and base area of 𝟐𝟓 𝟑cm2. Round to the nearest tenth, if necessary.

L = Ph

= 30(20) = 600 ft2

S = Ph + 2B

P = 3(10) = 30 cm

The base area is

Ex 1c. Find the lateral area and surface area of a cube with edge length 8 cm.

L = Ph

= 32(8) = 256 cm2

S = Ph + 2B

= 256 + 2(8)(8) = 384 cm2

P = 4(8) = 32 cm

The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.

III. Right Cylinders

Ex 2a. Find the lateral area and surface area of the right cylinder. Give your answers in terms of .

L = 2rh = 2(8)(10) = 160 in2

The radius is half the diameter, or 8 ft.

S = L + 2r2 = 160 + 2(8)2

= 288 in2

Ex 2b: Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of .

Step 1 Use the circumference to find the radius.

C = 2r Circumference of a circle

24 = 2r Substitute 24 for C.

r = 12 Divide both sides by 2.

Step 2 Use the radius to find the lateral area and surface area. The height is half the radius, or 6 cm.

L = 2rh = 2(12)(6) = 144 cm2

S = L + 2r2 = 144 + 2(12)2

= 432 in2

Lateral area

Surface area

Ex 2c. Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of .

Ex 2d. Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius.

Step 1 Use the circumference to find the radius.

A = r2

49 = r2

r = 7

Area of a circle

Substitute 49 for A.

Divide both sides by and take the square root.

Step 2 Use the radius to find the lateral area and surface area. The height is twice the radius, or 14 cm.

L = 2rh = 2(7)(14)=196 in2

S = L + 2r2 = 196 + 2(7)2 =294 in2

Lateral area

Surface area

Ex 2e. Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius.